
A field transformation, e.g. (REF ), will be a symmetry of the QFT if any correlator \(\langle X\rangle \)  remains invariant under that transformation, i.e. if [1]}:
\(\langle X\rangle &={\langle X\rangle }^\prime \\\text{where\hspace{14.22636pt}}{\langle X\rangle }^\prime \equiv \langle {\mathcal {T}}\tilde{\Phi }_1(\mathbf {x_1})\tilde{\Phi }_2(\mathbf {x_2})...\tilde{\Phi }_n(\mathbf {x_n})\rangle &:=\frac{\int [\mathcal {D}\mathbf {\tilde{\Phi }}]\text{ }\tilde{\Phi }_1(\mathbf {x_1})\tilde{\Phi }_2(\mathbf {x_2})...\tilde{\Phi }_n(\mathbf {x_n})\text{ }e^{iS^\prime [\mathbf {\tilde{\Phi }}]}}{\int [\mathcal {D}\mathbf {\tilde{\Phi }}]\text{ }e^{iS^\prime [\mathbf {\tilde{\Phi }}]}}\nonumber \) 
